
theorem
  for L being RelStr, A being Subset-Family of L st
  for X being Subset of L st X in A holds X is lower
  holds union A is lower Subset of L
proof
  let L be RelStr, A be Subset-Family of L such that
A1: for X being Subset of L st X in A holds X is lower;
  reconsider A as Subset-Family of L;
  reconsider X = union A as Subset of L;
  X is lower
  proof
    let x,y be Element of L;
    assume x in X;
    then consider Y being set such that
A2: x in Y and
A3: Y in A by TARSKI:def 4;
    reconsider Y as Subset of L by A3;
A4: Y is lower by A1,A3;
    assume y <= x;
    then y in Y by A2,A4;
    hence thesis by A3,TARSKI:def 4;
  end;
  hence thesis;
end;
